latent viral marketing, concepts and control methods
TRANSCRIPT
Latent Viral Marketing, Concepts and Control Methods Alon Sela Department of Industrial
Engineering
Tel Aviv University
Dmitri Goldenberg Department of Industrial
Engineering,
Tel Aviv University
Irad Ben-Gal Department of Industrial
Engineering,
Tel Aviv University
Erez Shmueli Department of Industrial
Engineering,
Tel Aviv University
Abstract โ Numerus works that study the spread of information in social networks include a
spreading mechanism in which a set of nodes is initially infected (i.e. seeded), followed by a viral
process, which spontaneously spread the message through the nodes of the network. These models are
used to describe the spread of rumors as well as the spread of new products and services. In reality
however, it is quite rare that a product or service spreads through a social networks solely by viral
forces. It is more common, that the spreader would invests a continuous active effort of its sales
representatives in order to enhance the spread. The Latent Viral Marketing Model is a spreading
model that fits this reality. Along the description of the model, the paper continues and proposes a
simple Scheduling of Seeding heuristics, which recommends the node to seed at each period. A large
study of empirical simulations shows that under a wide range of realistic initial conditions, the
Scheduling Seeding method improves a product adoption rate by 29%-94% in comparison to existing
state-of-the-art seeding methods.
KeywordsโInformation Spread; Social Networks; Information Cascades; Scheduled Seeding, Viral Marketing,
Linear Threshold.
I. INTRODUCTION
Social networks provide a powerful communication platform that is known to significantly influence social
and historical events. For example, its impact can be observed in President Obama`s first elections [1], [2],
in the Arab Spring uprising [3], in the spread of propaganda by ISIS recruiters [4], or in the US election of
2017. The importance of social networks is mainly a result of their efficiency in quickly spreading
information among many users. These traits are the reasons why political parties, terror organizations and
commercial firms, use social networks to a growing degree.
Generally, it is believed that information spread follow the Linear Threshold Model [5]. According to
this model, first, the spreader selects several chosen nodes and seeds them (where the act of a seeding [6],
reflects an intentional infection of nodes). Then, a viral process begins, where the information spreads
through the nodes of the social network and users infect each otherโs. Such an act of infection, can be
performed for example, if a user writes on his Facebook wall a new message, which is later seen by the
user`s friends. Then, the user`s friends can chose to send this message to their own friends. In each step. A
user can alternatively send the message directly to a single friend or send it to a group of friend. In Twitter,
the user spreads a message by Retweeting a received message, or simply by twitting a new tweet that
contains a relevant link. The follower of the user would then have this message presented in their tweets
time line, and can open the link. If they find the message interesting, they can retweet it, thus the message
will appear on the time line of their followers.
The spreading methods mentioned above are all active methods. This implies that users must perform an
action (such as a retweet or post on a wall) in order to spread the message to another. In each step, users
invest an effort (i.e. โworkโ) toward spreading the message. Such a method of information sharing is
desirable to many commercial firms. These firms seek to harness the viral forces of these many โfree
spreadersโ and have these spreaders invest their effort to spread the commercial firm`s products or services.
The firms wish to gain an almost unlimited source of free workers that spread their product or services.
Unfortunately, very few products spread solely by viral forces, and most firms still need to employ sales and
marketing departments to promote their products and services. The low ability spread products and services
by viral mechanisms is not due to the low importance of social forces in the act of purchasing. In fact, one`s
social connections are known for many years to have an immense influence on one`s personal decisions
making. The first social psychologists, Asch [7], Milgram [8], Granovetter [9] and Zimbardo [10], revealed
the importance of social influence as a key factor influencing one`s attitudes and values. Social proximity in
a social network predict tendencies that were believed to be genetic. For example, the tendency for obesity,
smoking, or even the tendency of being happy [11]. If such internal traits spread through the links of the
social network, shouldn`t we expect that recommendation for a new products would spread as well?
In practice, encouraging customers to invest efforts to spread a commercial product is not always as easy
as it seems [12]. First, it is required that these customers would like the product a lot! This is a critical
requirement in any effort for successful viral marketing. This is true, since an influential customer, which
actually dislikes the product, will influence the spread in a negatively [13]. Furthermore, customers do not
usually like to promote commercial firms by their own good will. A few relatively recent works [14], [15],
[16], [17], and [18] have shown that the tendency that a customer will spread a commercial products is lower
than previously believed [19], [20], [21]. These works observed the lengths of information cascades in large
data sets and found them rather short and shallow. It seems as the vast majority of messages never spread
through thousands of users, but rather through a relatively small number of users. Since large information
cascades are rare, it is also rare that a product or service spread solely by viral forces. An external aid to
spread is usually required for most products or services.
The following work proposes a model of information spread, which considers social influence in a more
realistic way. The method recognizes the importance of social forces, but does not expect to gain โfree
workersโ from it. An expectation to have a customer actively spreading a commercial product or services is
not too realistic. The customers influence on his friends is important, but the influence is not sufficient and is
not active, it is hidden, and latent.
As a motivating example, let us consider a setting in which a given company aims to promote the sales
of one of its products. The companyโs sales representatives might contact customers and offer them to
purchase the product by phone, or by using an equivalent advertising platform. If a customer purchases the
product, he / she might tell some of his / her friends (network neighbors) about this purchase. We assume
that these friends will not actively contact the company to purchase the product by themselves, but would
rather keep the positive recommendation latent in their minds. However, if contacted by the companyโs sales
representatives within a certain period, the positive recommendations accumulated on the product, as
provided to them by their friends, is likely to influence their likelihood to purchase the product. If the sales
representative address the customer long after the customer`s friends` recommendations have been heard, the
customer is less likely to purchase. The company thus needs to decide which users to approach and at what
points in time in order to utilize its sales budget in an efficient manner, while taking into account the latent
influence; e.g. the effect of the user`s friends recommendations as is accumulated in the customer`s minds.
This work fits the scenarios above. We define the Latent Viral Marketing Model (LVM), and a related
seeding method, the Scheduling Seeding Heuristics (SSH), which increases the number of successful seeding
attempts in the above scenario. The work adds a stochastic aspect to the deterministic Scheduled Seeding
method that were developed and presented in our previous works [22], [23], and adds the realistic latency to
the sequential seeding process described in [24]. Thus, the work follows a setting that we believe fits real
life scenarios to a greater degree.
According to a large set of simulations, SSH significantly improves the number of successful seeding
attempts in scenarios similar to the LVM model described above, in comparison to existing state-of-the-art
seeding benchmark approaches.
These benchmark approaches mainly focus on careful selection of nodes with high centrality measure
[25], such as PageRank, Eigenvector Centrality, or simply the nodeโs Degree Rank in the initial seeding
process. More specifically, the SSH reaches an average improvement rate of 23%-153% in the number
successful seeding attempts (depending on initial conditions), and in some extreme cases reaches an
improvement of up to 10 folds.
The next section includes a brief background on information diffusion models through social networks
and in particularly, on the Linear Threshold model. The background section is followed by an in-depth
description of the proposed LVM information spread model followed by the SSH seed selection heuristics.
We present the results of various simulations experiments and summarize the study by a concluding
paragraph.
II. MOTIVATION
A. The Linear Threshold Model
One of the most popular models in the field of viral marketing is the Linear Threshold model [5]. This
model starts when an initial set of nodes is first infected, followed by a viral process model which simplifies
social influence. According to the Linear Threshold model, the viral spread will flow if โ ๐๐ฃ,๐ค โฅ ๐๐ฃ๐คโ๐๐ฃ,
where ๐๐ฃ denotes the set of infected neighbors of ๐ฃ, and ๐๐ฃ,๐ค denote the weights; i.e. the social influence
that ๐ค activates on ๐ฃ. If the total influence reaches a threshold ๐๐ฃ, node ๐ฃ changes its state and becomes
infected.
Plotting the total number of infected nodes versus the elapsed time, while applying the Linear Threshold
viral process, creates a plot that often resembles a sigmoid function. The number of infected nodes slowly
increases at the beginning of the process, then after enough infected nodes accumulate; it increases sharply,
up to the point where most nodes are infected. Then, when additional uninfected node becomes scarce, the
speed of infection slows down, and the slope decreases.
Similar sigmoid spreading curves represent many physical phenomena of spreads, such as for example a
forest fire or virus epidemic. In a forest fire, after an intentional ignite (i.e. seeding) the fire spread is first
slow. Then, as the fire grows, it quickly spreads by its own forces to the rest of the forest. In this period, in
many cases, the fire can burn large parts of the forest in a short period. At the end, when much of the forest
is already burn, the fire slowly decays since unburnt trees are infrequent, until it completely vanishes.
This sigmoid growth function, while fitting numerus natural spreading phenomena, does not seem to fit
the spread of ideas through social networks.
B. Growth of Actual Information Cascades
A growing body of works, [15], [17], [26], [27], [28], which analyzed several large social networks data
sets, claim that large information cascades are rater rare. Most information cascades only spread through two
people, even fewer spread to three. For example, a spread of a message to five friends occurs in only 1/8 of
the messages, and a spread of a message 3 times in a vertical cascade (i.e. an initial message that is spread to
a friend, and this friend spreads it again to his/her friends), only occurs in 1/16 of the messages. An even
larger spread, for example a vertical spread of 8 steps, was found to only occur in between 0.01% - 0.001%
of the messages. These studies were replicated over different social networks; and capture an important
aspect of information cascades. While people collect information consistently, they do not always actively
diffuse it further to their friends.
Information cascades differ from the spread of biological virus in their selective nature. While collecting
the information might be similar to receiving a virus, and people do collect much of the information they
receive, information spread is more selective. People tend to distribute information cautiously and not repeat
everything that they have heard to everyone.
This is one of the reasons, why it is rather rare that a company succeeds to distribute its products with no
additional effort, simply by using a viral process. In contrast to a virally based strategy, most companies
need to spend a lot of effort (and budget) to actively help the spread of their products. Most companies need
to construct brands names through commercial communication methods, they employ sales personnel, and
they actively promote social network marketing strategy. The conventional Linear Threshold model does not
address the every dayโs scenario, in which a company invests substantial effort to promote a product or a
service. This is why the LVM model is required.
C. Assumptions Underlying the Latent Viral Marketing Model
The first works on information spreading through social networks compared the spread phenomena to
the spread of viruses. The SIR model is the basic model of virus spread. It has been studied for over a
hundred years, with an excellent work summarizing the vast studies in the field found in [29]. Unlike the
spread of a biological virus, social norms influence the adoption of ideas. The rate of acceptance of a certain
idea in one`s social circle predicts the likelihood for adopting it. Social norms are indeed integrated in the
Linear Threshold model, which defines the probability of infection as the sum of intentioned neighbors`
weights. Although the theoretical importance of this work, along similar [30], [31] information cascades
works is evident, these works does not fit a case of commercial products or services in which the spread
requires continues effort of marketing and sales departments.
In order to fit the Linear Threshold model to these scenarios, we must first change the deterministic
nature of the model. Another required modification is the clear separation between the seeding stage and the
viral stage, which does not fit the reality of a commercial product spread. The investment of an entire budget
in a single and initial period is in many cases impossible. Most commercial firms have limited call centers
capacities, and can only reach a limited number of customers per day. Lastly, many spreading models
assume that if a certain number of neighbors of a person adopt a product, the person will adopt it as well.
While this might be true, in many cases one might be willing to adopt a product or service following good
recommendations from friends, but he does not adopt it simply because he is too busy to actively reach the
company and acquire the product or service. Nevertheless, if reached by a sales person, he is likely to adopt
the product or service.
The Latent Viral Model provides a new framework, which we hope overpass the obstacles mentioned
above. It assumes from one hand that nodes accumulate social information, and that this accumulated
information is a major factor in the adoption decision. However, in contrast to the previous Independent
Cascade philosophy, it assumes that new nodes cannot become infected solely by a viral process. Instead, an
external effort of a sales representative is required before a node actually becomes infected. Thus, the
question of seeds allocation is relevant not only in the initial stage, but also along the entire spread process.
The work follows previous works [22], [23], [32] on the topic, and studies the stochastic aspect of the
Scheduling Seeding problem. It includes a deeper inspection of the ideas presented in these previous studies.
Correspondingly, the challenge is to decide on what node it is worthy to invest the seed and at what
period. This decision is based on the feedbacks received from previous seed attempts along the current social
network structure. Such feedbacks include the knowledge of customers that have already adopted the
product, along those who have not. As shown in the result section, when using the LVM model, the success
rates of the seeds trials grow if the Scheduling Seeding Heuristic (SSH) is used. A more formal description
of the model followed by the heuristics used to select the seeds is presented in the next section.
III. THE PROPOSED MODEL
A. The Setting
Let us consider a company with good visibility on the social network of its clients. The company wants
to offer its customers a new service or product (we use the term service or product interchangeably), and
offers it through its sales representatives. The company seeks to achieve the highest possible number of
customers that adopt the new service, and allocates a limited budget, denoted by ๐ต, to promote this goal.
If the company offers the service to a customer; let us say ๐ฃ, the customer might accept or reject the offer
with a certain probability ๐. This probability is affected by the adoption rate of the service by the customer`s
social circle, as further explained below in eq.(2). In a case where the customer refuses the offer, subsequent
offers in the near future will only annoy the customer, and therefore the product would not be offered again
to the same customer. In such a case the customer`s state is considered to be in a โSeeding Failedโ state. The
social influence is such that if a customer accepts the offer, the customer is likely to influence his / her
friends for the next ๐ก๐๐๐ periods. In this case, after ๐ก๐๐๐periods have ended, he the customer changes from a
state of being infected and infectious, to a state of being infected but not infectious. This state change
reflects the retention loss, or the loss of interest in the message due to information overload [33]. The
possible states of a customer are denote by ๐๐ก๐ฃ and as defined below.
(1) ๐บ๐๐ = {
๐ โ ๐ต๐๐ โ ๐ฐ๐๐๐๐๐๐๐ ๐ โ ๐ฐ๐๐๐๐๐๐ ๐๐๐ ๐ฐ๐๐๐๐๐๐๐๐๐ ๐ โ ๐ฐ๐๐๐๐๐๐๐ , ๐ต๐๐ โ ๐ฐ๐๐๐๐๐๐๐๐๐ ๐ โ ๐บ๐๐๐ ๐๐๐ ๐ญ๐๐๐๐๐
The various states changes ๐๐ก๐ฃ that customer ๐ฃ might follow, are presented in the states transition scheme in
Fig. 1.
Fig. 1. Possible States and the causes for states changes of a customer.
B. Defining the Probability of a Successful Seed Attempt
The probability that customer ๐ฃ accepts an offer is affected by the social pressure executed on the customer,
as well as the attractiveness of the proposed product or service itself. We therefore define a maximal
probability for an infection (adoption) by the proposed service or product, and denote it by ๐๐ด๐๐๐. This
parameter depends on the type of product or service, and can usually be estimated from past data. For
example, the probability of accepting an offer for three months free cable TV service without any
commitment might be rather high, while the probability of accepting an online purchase of a new luxury car
is low. The probability of accepting the proposed offer follows eq.(2), where ๐๐ฃ+ is the set of indexes of the
infected (๐๐ก๐ฃ = 1) neighbors of customer ๐ฃ, and ๐๐ฃ is the minimal number of infected neighbors at ๐๐ด๐๐๐.
This formulation fits the results appearing in Asch`s conformity experiments [7], [34], which inspected the
probability of conforming to norms as a factor of group size (see Fig.2).
(2) ๐๐ = ๐๐ด๐๐๐ฃ(
๐๐๐ (๐ฝ๐, |๐ต๐+|)
๐ฝ๐ )
In his works, Asch inspected how group size influences the probability of conforming to the opinion of
the majority. As the coalition of this majority grows, the conforming probability grows almost linearly, until
a certain coalition size (see Fig.2 right), where we denote this value by ๐๐ฃ. Note that larger coalitions
above ๐๐ฃ will not increase the likelihood of conforming any more. This influence function is plot in Fig.2.
(left) which presents the probability of conforming as it was copied from Ash`s original article on social
conformity [34]. In comparison, the same right image of the figure presents the approximated function as
defined in eq.(2). In both figures, the x-axis represents the number of people adopting the opinion, and the y-
axis indicates the equivalent probability of acceptance, with the peak y-axis equal to ๐๐ด๐๐๐.
Fig.2. Social Influence Function based on Aschโs conformity experiment. Right, figure directly copied from Asch`s article [34], as
compared to left figure which presents an approximation function as defined by eq.(2), with parameters ๐๐ด๐๐๐ฃ= ๐. ๐๐ and ๐ฝ๐=4.
C. The Pre-Seeding and Seeding Processes
According to the definition of the acceptance probability as defined in eq.(2), if there is not even one
infected node in the entire social network graph ๐ฎ = (๐ฝ, ๐ฌ), the term|๐ต๐+| = 0; โ๐ฃ โ ๐ฝ. It follows that ๐๐ฃ =
0; โ๐ฃ โ ๐ฝ, and if course, in such a case, no seed trial would succeed. To prevent of being trapped in such a
zero attractor, prior to the spread process, we define an initial set of infected nodes and set them to
state ๐บ๐๐ = 1; i.e. infected and infectious. These nodes are chosen randomly from ๐ฝ, and this pre-seeding
infected set is defined by ๐ญ๐๐๐๐. The relative size of |๐ญ๐๐๐๐| is usually small, and consists of less than 1% of
the nodes. Furthermore, the infection times of the nodes in ๐ญ๐๐๐๐ are set such that each of these nodes has a
different initial infection time, thus they do not change from ๐๐ก๐ฃ = 1 โ ๐๐ก๐ฃ = 2 at once but rather gradually.
Following the initial setting of ๐ญ๐๐๐๐, the seeding process starts. The process includes ๐ต seeds attempts,
which are performed on selected nodes. Assuming each seeding attempts costs exactly one unit of budget,
and ๐๐ nodes can only be seed at each period, these limitation fits real scenarios in which call centers can
only make a limited number of phone calls per day due to their work hours constraints.
The Scheduling seeding algorithm nodes selection include three steps. First, the algorithm chooses a set
of nodes that are not yet infected, but have at least one infected neighbor. These are the potential candidates
for the seeds. Second, it computes an โattractivenessโ score for each of these candidates. Third, in each
period, ๐๐ nodes with the highest scores are seeded.
After the seeding is performed, the simulative stage โdecidesโ if the seeds are accepted or rejected. The
seeding trial succeeds or fails with a probability ๐ , which as defined according to eq.(2). After the seeding of
each period, relevant parameters and state changes are executed for the relevant nodes in the network. These
include the calculation of ๐ต๐+ for each node, as well as changes of states for odes that require such a change.
This process ends when the entire budget is depleted, or when all the nodes in the network becomes infected.
Once the process ends, the ratio of successful seeding is computed, simply as the number of seeding
successes per seeding trials.
D. The Seeding Scheduling Heuristics (SSH) Score Computation.
The Seeding Scheduling heuristics recommended the seed trials by computing their LVM score at each
step. The score is based on the expected value of the node being seeded, and reflects the probability of an
occurrence of an event ๐(๐), multiplied by the utility ๐(๐) of the event. The event ๐, is defined as the
success in a seed of node ๐ฃ.
The utility function gained from ๐ is constructed from two separate parts. First, the success seeding of ๐ฃ
has a utility of one additional infection node. Second, to this term, an additional term is added as the utility
gained from the increased probabilities of future successful seeding of the uninfected neighbors of ๐ฃ. Since ๐ฃ
is now infected, its neighbors are now easier to seed. The first term, the utility gained from the successful
seeding of ๐ฃ simply equal to 1. The second term; the increased probabilities of uninfected neighbors of ๐ฃ is
defined as the sum of changes over all nodes ๐ข โ ๐๐ฃโ where ๐๐ฃ
โ denote the non-infected neighbors of ๐ฃ.
Assuming the event ๐ occurs, this second term is the value of the nodes ๐๐ฃโ: ๐ minus their current
value ๐๐ฃโ: ๏ฟฝฬ ๏ฟฝ. Thus, the utility from seeding trial to ๐ฃ is ๐(๐ฃ) = โ โ ๐(๐) โ๐ขโ๐๐ฃ
โ ๐(๐) - โ โ ๐(๏ฟฝฬ ๏ฟฝ) โ๐ขโ๐๐ฃโ ๐(๏ฟฝฬ ๏ฟฝ),
where ๐ is the states of the neighboring nodes ๐ข โ ๐๐ฃโafter the seed of ๐ฃ succeeded, and ๐ ฬ is their states
before the seed of ๐ฃ succeeded. This is a recursive formulation, since ๐(๏ฟฝฬ ๏ฟฝ) is actually unknown.
The computation method to calculate this score, is performed recursively, and is defined for a depth of 3
recursion levels in eq.(3). The recursive computation of the score, for a depth of k iterations, is presented in
the following algorithm, and is later named as the seeding strategies โpicky_social_<k>โ; i.e.
โpicky_social_0โ, โpicky_social_1โ and โpicky_social_2โ, where picky_social_0 computes the score simply
as the term ๐(๐ฃ), picky_social_1 as ๐(๐ฃ) โ {1 + โ [๐(๐ข)]}๐ขโ๐โ(๐ฃ) and picky_social_2 as the full in eq.(3).
(3) ๐๐๐๐๐(๐ฃ) = ๐(๐ฃ) โ {1 + โ [๐(๐ข) โ (1 + โ ๐(๐ค)] ๐คโ๐โ(๐ข) }๐ขโ๐โ(๐ฃ)
This attractiveness score is computed recursively as defined in the pseudo-code below
The SSH Scoring Algorithm
Function Social Score(v,G,k): # input: v - relevant node, G - Graph, k - Levels (Social_0, Social_1, Social_2)
๐๐๐ก ๐(๐ฃ) = ๐๐ด๐๐๐(
๐๐๐ (๐๐ฃ, |๐๐ฃ+|)
๐๐ฃ ) # Probability of infection of v in current time step
๐๐ ๐ฟ๐๐ฃ๐๐๐ = 0: ๐๐๐ก๐ข๐๐ ๐(๐ฃ) # Level 0 - Greedy score ๐๐๐ ๐: ๐ ๐๐ก ๐๐๐๐๐ = 1 ๐๐๐ ๐ข ๐๐ ๐โ(๐ฃ) # go over all vโฒs non โ infected neighbors ๐ ๐๐๐๐ = ๐ ๐๐๐๐ + ๐๐๐๐๐๐(๐ข, ๐บ, ๐ฟ๐๐ฃ๐๐๐ โ 1) ๐๐๐ก๐ข๐๐ ๐(๐ฃ) โ ๐๐๐๐๐
To clarify the above recursive method, note that the expected value of v itself is p(v)*1, and the expected
value of u (where v is the 1st circle) which weren't infected is p(v)*p(u|v)*1, meaning an occurrence of both
events (successful infection of v and afterward successful infection of u which is based on a new probability
calculation. Similarly, the expected value of w (where v is in the 2nd circle, and u is in the 1st circle, is :
p(v)*p(u|v)*p(w|u,v)*1), which is the formulation defined in eq.(3).
While the above function computes the scores at any level of k, as further seen in the result section, there
is a tradeoff between the effort to foresee and the time of computation. In most cases, it seems as the right
balance is in one single level of depth, that is, in setting the parameter k=1. At this depth of recursive, the
results are rather good, but the additional computation complexity dramatically increases. In the next
section, we present the methods used to evaluate the efficiency of the above SSH scorings under the LVM
modeling, followed by the results from these sets of experiments.
IV. EVALUATION
A. Experimental setup
We set an empirical experiment in order to compare the performance of suggested and existing
benchmark seeding heuristic. The experiment was executed on a Linux Cluster with 128 GB RAM memory,
where the simulations were run on a single Intel 2.7GHz CPU. Each simulation instance started with a setup
of the initial condition, which included a selection of a pre-simulation infected set ๐ญ๐๐๐๐ as defined above.
The infected time of this set were generated from a uniform distribution, such that there would not be a
sharp decline in the infectious nodes at period ๐ก = ๐ก๐๐๐ + 1.
As the simulation started, in each period, a seed was offered to a single node, whereas the selection of the
seeded node was based on different heuristic rules. Each seeded node could accept or reject the seed with a
probability based on its surrounding nodes according to eq.(2), and the node`s state function changes were
calculate at each discreet period. The simulation instance ended when the entire budget was used, then, the
final seed success to failure rate was calculated.
The results examine the different seeding strategies across changing dimensions of initial parameters,
such that at each set of simulations, a single dimension was examined across a wide range of values. The
other parameters were set to their default values, which were in most cases the median of the range. During
each simulation run, the SSH seeding recommendations under the LVM simulations was compared to the
benchmark seeding methods, throughout the entire parameter space, while running each parameter
combination for at least 400 replications. The entire parameter space as used in the simulation experiment is
presented in Table 1 below.
Table 1 โ Simulation Parameter Space1
Parameter Values
Network (see Table 2 for further
details).
Sampled Citation network, Slashdot Network,
Sampled EuEmail network, WikiVote Network,
Epinions Network, Enron Network
Network size (sample # nodes) 5000, 10000, 50000, 100000, 500000, 1000000
Initially infected population size 50, 100, 200, 500, 1000
Max Budget 50, 100, 200, 500, 1000
Threshold 3, 4, 5, 6, 7
Maximal Probability 0.1, 0.3, 0.5, 0.7, 0.9
Infection Time (Time of Oblivion) 10, 20, 50, 100, 200
Seeding method (our method and the
benchmark methods)
Random, GEC, Picky Random, Picky_GEC, LVM
(Picky_Social_0, Picky_0ocial_1, Picky_0ocial_2)
Three different SSH seeding recommendations under the LVM simulations scheme i.e. Picky_Social_0,
Picky_social_1, Picky_social_2 were compared to four benchmark methods which included (1) Random, (2)
GEC, (3) Picky Random, (4) Picky GEC as further defined in the next section. These simulations were
executed on different networks as defined in Table 2 below.
1 Default value of each dimesion is marked with bold. The value is used when other parameters change.
Table 2 โ Networks2 Used in Simulation
Network Number of Nodes Avg Degree Avg Clustering Network Type
Citations 1000000 2.83481 0.039113922 Sampled & Undirected
Citations 500000 4.057372 0.060063242 Sampled & Undirected
Citations 100000 7.60482 0.136068811 Sampled & Undirected
Citations 50000 8.19712 0.160465584 Sampled & Undirected
Citations 10000 6.809 0.200986075 Sampled & Undirected
Enron 36692 10.020222 0.49698256 Full & Undirected
Wiki_vote 7115 28.323823 0.140897846 Full & Undirected
Slashdot 82168 14.179072 0.06034486 Full & Undirected
Euemail 100000 1.56686 0.034104364 Sampled & Undirected
Epinions 75879 10.694395 0.137756373 Full & Undirected
B. The Benchmark Methods
The evaluation of the SSH seed recommendations method for the LVM scheme, was compared to the four
benchmark methods below.
(1) Random โ Randomly choosing one uninfected node to seed at each time step.
(2) GEC โ Choosing the one uninfected node with the highest Eigenvalue Centrality measure at each time
step.
(3) Picky Random โ Choosing a random uninfected node from the nodes which have at least one infected
and infectious neighbor.
(4) Picky GEC โ Choosing an uninfected node with the highest Eigenvector centrality from the nodes that
have at least one infected and infectious neighbor.
These benchmark methods were compared to these three SSH social heuristics.
(1) Picky_Social_0 - Choosing a non-infected node with the highest value of ๐(๐ฃ) at each time step, see
SSH Scoring Algorithm with k=0.
(2) Picky_Social_1 - Choosing the non-infected node having the highest value of Score(v) as defined in
the first part of eq. (3), and the SSH Scoring Algorithm with k=1.
(3) Picky_Social_2 โ Choosing the node having the highest value of Score(v) as defined in eq. (3) and the
SSH Scoring Algorithm with k=2.
In the simulations, we first assume that the values of the parameters ๐๐ฃ and ๐๐ด๐๐๐ are known. In the second
set of experiments, we assumes that we only know the mean and variance of these parameters, along their
distribution. The means are denoted by ๐๐ and ฮผ ๐๐ด๐๐๐, and the variance are denoted by ฯฮธ and
ฯ ๐๐ด๐๐๐accordingly. The real value of these parameters for each node were generated prior to each
simulation run, and were not preliminarily known to the SSH recommendation algorithm.
V. RESULTS
A. Centrality of the nodes chosen to be seeded
The Eigenvector Centrality measure of a node (as well as its PageRank score) are considered as a good
proxy for a node`s ability to spread information. The main concept behind the LVM Scheduling scheme is
2 Networks were downloaded from SNAP, http://snap.stanford.edu/data/index.html then sampled.
that it is not only the centrality of the node that defines its importance, but rather its tendency to accept the
information at any precise period of time.
Before presenting the entire sets of results, we thus first examine the nodes chosen for the seeding at
each period in regards of their Eigenvector centrality. This inspection allows us to validate that the success
of the SSH method is not simply because it prefers seeding central nodes.
We compare the Eigenvector Centrality of each seeded node along time, when using the GEC method, a
method that allocates the seed to the relevant non-infected nodes by their Eigenvector Centrality scores, to
the scheduling methods by the LVM method (i.e. named Social).
A comparison presenting the centrality of the selected nodes in the LVM method is presented in Fig.
3Fig. 3. below. The SSH Social method, (blue line) allocates seeds to nodes with relatively lower average
Eigenvalue Centrality, as compared to the GEC method. While in a single run (presented in the interior plot
of the figure), we can see that nodes with high Eigenvalue Centrality can be seeded in a rather later stages,
the average Eigenvalue Centrality (presented in the exterior plot) of the nodes along time is substantially
lower for the SSH.
It can be concluded from those first results that the LVM method does not allocate the seeds to central
nodes, but rather that it allocates them to nodes that are of high importance at any current point in time. As is
further seen, the SSH selection of nodes does not only allocate seeds to less central nodes (which in reality
might be easier to reach), but rather results in final success rate that is substantially higher that of the
benchmark methods for any given budget.
Fig. 3. Eigenvector Centrality of the nodes chosen for seeding. Allocation of seeds to influential nodes; i.e. highest Eigenvector
Centrality (orange), as compared to the allocation of seeds by the SSH social methods (blue). The x-axis is the time of seed
attempt, while the y-axis is the Eigenvector Centrality of the node on which the seeding attempt is performed.
B. Comparing the LVM with the Benchmark methods
We start by comparing the SSH method to the benchmark methods, for different network sizes. As can be
seen in Fig. 4 below, the social methods (blue bars) outperform the benchmarks methods by almost twice.
For all the different seeding methods, the Social 2 method seem to reach the best results, followed by the
social 1 and the Social 0 methods. In comparison, the benchmark method of Picky GEC, a method that
allocates seeds to nodes with the highest Eigenvector Centrality in condition that these nodes already have at
least one infected neighbor, only succeeds at about ~13% as compared to ~20% success rates for the Social
method. Note that the GEC and the Random methods practically used by many commercial firms that do not
include the network structure of their clients in their marketing efforts. The success rates in these methods
are far lower.
Fig. 4. Comparison of SSH Scheduling method (blue scale) to the benchmark methods for different sizes of networks.
The results in Fig. 4 are on sampled citation networks of different sizes. We follow these results and validate
them for different networks, on diverse average degrees and Clustering Coefficients.
Fig. 5. Comparison of SSH Scheduling method (blue scale) to the benchmark methods for different networks topologies.
As seen in Fig. 5, the results are mainly similar. Note that the euemail-100000 network has substantially
lower success rates as compared to the other networks. To understand these results, we need to look at the
average degree of this network and compare it to the average degree of the other networks (see Table 2).
While euemail-100000 network has an average degree of 1.56, the other networks have an average degree of
10.7 or higher. The low degree in the euemail-100000 network reduces the probability of any seed success,
since in the LVM model, the number of infected neighbors is a major factor influencing the probability of a
successful seed, when the network is sparse, and this probability is accordingly low.
Note also that in these results, unlike the case of the generated network, the Social 2 is not always the
best method. Similarly, Social 1 is not always better than Social 0. It seems as in reality, when the network
topologies differ, in many cases it is better to use the simple Social 0 and Social 1 heuristics over the more
complex Social 2 heuristics which tries to plan forward for two steps in advance.
The GEC methods seed nodes with high Eigenvector Centrality in earlier stages. This might create a larger
influence at early stages and improve the acceptance rates later on. In order to inspect the temporal aspect of
the spread, we measured the success rates of the different seeding methods along the time axis. These
results, as presented in Fig. 6, indicate a growth in the success rate along the time axis. The growth is larger
in the Social seeding methods (blue), as compared to the non-social methods (yellow or green). The growth
in the success ratio seems to follow a log like function, since the y-axis is the ratio and not the absolute
number. These results imply that for growing budgets (growth in time) we expect a constant benefit from
using the social methods as compared to the benchmark methods.
Fig. 6. Temporal comparison of the aspect of the SSH Scheduling method (blue scale) to the benchmark methods
As described in the proposed model section, prior to the seeding attempts, the states of nodes in ๐น๐๐๐๐ก
were set to ๐๐ก๐ฃ = 1. We inspect the influence of the size of ๐น๐๐๐๐ก on the different seeding methods.
As demonstrated in Fig. 7. below, a larger initial population in ๐น๐๐๐๐ก (x-axis) improves the relative utility
of the Social methods. When the initial set ๐น๐๐๐๐กconsists of only 50 infected nodes, the Social methods
succeeds in the seeding 16.6%-18.4% of the seeding attempts. In comparison, the Picky GEC methods
succeeded in the seeding of 11.5% and the Picky Random only succeeds in 9% of the cases. This is an
improvement of 44% for the Social methods. As opposed to this initial setting of ๐น๐๐๐๐ก , if ๐น๐๐๐๐ก = 1000, the
social methods succeeds in seeding 29%-28.1% of the seeding attempts, while the Picky GEC and Picky
Random only succeeded in 14.5% and 9.1% which is an improvement of 94%. Thus, the improvement of
the Social methods over the next best methods grow from 44% to 94% as ๐น๐๐๐๐ก grows.
Fig. 7. Influence of number of infected nodes prior the seeding on the different Scheduling method. Social marked by blue scale, as
compared to GEC and Random methods marked Yellow and Green accordingly.
C. Initial Parameter Estimations
In the LVM model, the probability of a successful seeding, as defined in eq.(2) is based on the three
parameters. The first is the number of adopters in the social circle of a node ๐ฃ; denoted |๐๐ฃ+|. This parameter
is known, assuming that we know exactly which of the node`s neighbors have adopted the service or product
that is being spread. The two other parameters are ๐๐๐๐ฅ and ๐๐ฃ which are unknown. We will first present an
analysis, which inspect the influence of these parameters on the results, followed by an inspection of the
more realistic scenario, in which the values of these parameters are not known, but they are estimated
through their statistical properties.
Fig. 8. Influence of ๐๐๐๐ฅ (left figure) and ๐๐ฃ - the threshold (right figure) on the success rates of the Social methods (marked blue lines)
compared to the benchmark methods (marked yellow and green lines).
The influence of ๐๐๐๐ฅ and ๐๐ฃ on the results can be observed in Fig. 8. It is clear that higher values of
๐๐๐๐ฅ (left image), only improves the efficiency of the Social methods (marked by blue lines) as compared
to the other benchmark methods (marked by green or yellow lines). This result make sense. A product or
service that have a large value of ๐๐๐๐ฅ are those that have a larger probability of purchase if one`s friends
have purchased. For example, such products can be trendy products for teenagers or kids, where the social
influence plays a large role in the desirability of the product. For these products, it would be reasonable to
assume that the LVM method, a strategy that better incorporates the social aspect of the purchasing decision
would be beneficial over more static approaches, which only include the network topology.
Regarding the threshold value ๐๐ฃ as presented in the right figure, higher values of ๐๐ฃ represent products
where one need to accumulate more adopting neighbors before one reaches a purchasing maturity. Products
or services that fit the category and are expected to have high values of ๐๐ฃ, are products or services where
one tends to accumulate much information prior the purchasing maturity. These can be important (and
costly) decisions such as buying a new car or new home. In these important decisions, where one tends to
invest one`s time and effort in profound inquiries prior the purchasing decision, the social aspect is less
dominant. While the trend seem to continue such that the social LVM methods are always preferred, these
are decisions where the success ratio is also small. Note that such cases as expensive decisions, the social
methods (when one consults as many as 7 friends) is 8%-10% for the Social methods, as compared to 6% for
the picky GEC method. This represent an improvement of at least 33% for the Social methods over the best
of the other methods, which in a case of an expensive product or service is a very good result.
D. Inspecting the Simulations Space with Unknown Parameters
The results described in the section above assumed that the values of ๐๐๐๐ฅ and ๐๐ฃ are known. Of course,
this is not true. At the best, the distribution of these parameters can be estimated, but the individual
parameter for each node is never known. For this reason, we conducted another set of experiments and
inspected the performance of our scheduling method under the LVM for unknown values of ๐๐๐๐ฅ๐ and ๐๐ฃ.
In these experiments, the means and standard deviations of ๐๐๐๐ฅ๐ and ๐๐ฃ were known, but the true value of
these parameters for each node was not revealed to the Scheduling algorithm.
We thus first generated value for ๐๐๐๐ฅ๐ and for ๐๐ฃ prior the run, then run the different seeding methods
while not letting the algorithm know the values of the parameters for each node. In each run, the Scheduling
algorithm simply generate a possible value for ๐๐๐๐ฅ ๐ and ๐๐ฃ from their means, standard deviations and
distributions and continued to search for the best node as if their values were known. We assumed the
parameters distribution is a Normal distribution, and inspected the influence of growing rates of errors rates
with growing standard deviations of these parameters.
Fig. 9. Social methods improvement rates for different degree of uncertainly
As seen in Fig. 9, the growing degree of uncertainty (x-axis) of the real values of ๐๐๐๐ฅ ๐ and ๐๐ฃ, results
in a decreasing performance of the social LVM methods, as compared to the random method. We set the
Picky Random method (which randomly selects nodes that have at least one infected neighbor) as a
comparison line, and only inspect the degree in which each distinct method performs better than the random
method. Note that for the random heuristics (inner plot); an addition of noise actually improves the
performance of the method. If the values of ๐๐๐๐ฅ๐ and ๐๐ฃ have larger variance, it implies that in some cases
๐๐๐๐ฅ ๐ and ๐๐ฃ would be low. In these cases, if the nodes selection is random, the probability of a seed
success is high. Since we compare the performance of each heuristics to this random heuristics, which grows
with the addition of more noise, we expect that more โnoiseโ to result in a less accurate plan of the Social
method as compared to the random method. It can be seen, that even for the high levels of a standard
deviation of as much as 2 standard deviations, the worst Social method (i.e. Social 0) still performs better
than the random by 153%, and the best benchmark method (i.e. Picky GEC) only performs better than the
random by 124%. This represents an improvement of at least 23% for the Social method over the best other
benchmark method. Furthermore, for smaller levels of standard deviations, (cases where we can better
estimate the parameter values) the improvement of the Social methods as compared the other methods is
substantially higher.
E. Aditional Unknown Parameters of Minimal Probability of Adoption
The previous section inspected the behavior of the model when the parameters ๐๐๐๐ฅ ๐ and ๐๐ฃ were
unknown. These parameters represent the uncertainty related to the highest possible probability of seed
success, in a case where there are many infected neighbors. There is nevertheless, another source of
uncertainty, which was not addressed in the LVM model. This is the case of a product adoption when none
of one`s friend have never adopted it. It is clear that while social influence is an important aspect in the
purchasing decision, there are cases where one purchase a product or service that none of one`s friend have
purchased.
As seen in eq.(2), when none of one`s friends have adopted the product or service, the value of |๐๐ฃ+| = 0,
and the probability of adoption is accordingly 0. This difficulty in the model, of course needs a correction. In
order to correct it, we redefined eq.(3), as below, by adding a minimal value ๐๐ด๐๐๐ฃ to the term.
(4) ๐๐ฃ
= ๐๐๐๐๐
+ (1 โ ๐๐๐๐๐
)[๐๐๐๐ฅ๐
(min (๐๐ฃ, |๐๐ฃ
+|)
๐๐ฃ )]
The term ๐๐ด๐๐๐ฃ thus represent the a priori probability of a node accepting a seed, when none of its
neighboring nodes have accepted it.
Fig. 10. Influence of ๐๐ด๐๐๐ฃon the success rates of the Social methods (marked blue lines) as compared to the benchmark methods.
The additional term ๐๐ด๐๐๐ฃ to the LVM model, reveals two interesting properties of the LVM model and
the Social heuristics. First, note that when ๐๐ด๐๐๐ฃ is added to the LVM model, the Social 0 method
outperform the other Social methods. This trend can be explained by the low ability of the more complex
Social Algorithm to correctly predict the seeds success when noise is added. Furthermore, note that when
the value of ๐๐ด๐๐๐ฃ is ๐๐ด๐๐๐ฃ
โค 0.4 the Social methods are still better than the other methods. In contrast,
when the value of ๐๐ด๐๐๐ฃ is ๐๐ด๐๐๐ฃ
> 0.4 the Picky GEC methods gains better results over the Social
methods.
These results define the region where the Social methods is expected to gain better results, and enable a
better decision when to use the Social methods and when to use the GEC methods. With this in mind, it is
important to note that the Picky GEC method does not simply allocate seeds to nodes according to their
Eigenvalue Centrality, but rather restricts the nodes allocations to nodes that have at least one infected
neighbor. It this includes some type of feedback on what node adopted the offer. If this feedback is ignored,
then the correct comparison needs to be the GEC regime and not in the Picky GEC regime. In this method,
seeds are allocated to nodes according to their Eigenvector Centrality without concerning their neighborsโ
state at all. In this case, only when 50% of the purchasing decision is personal ( ๐๐ด๐๐ > 0.5) it is better to
use the GEC methods over the Social methods.
F. Run Time of the Social Methods
The different Social methods represent a growing degrees of future planning effort. While the Social 0
method is fully greedy, the Social 1 tries to plan one step ahead, and the Social 2 plans two steps ahead.
Although the SSH scoring algorithm, as previously presented can be used with growing degrees of future
planning, we did not find sufficient improvement in more than 2 steps plan ahead. This is important if
considering the fact that when the networks size grow, the computational cost of the plan ahead grows
accordingly. Furthermore, in many cases, a trial to plan for the far future might result in trial to seed nodes
that are influential in the long term, but have lower probability of accepting the seed in the short term. Such
a strategy can result in lower final success rates since these influential nodes seeding simply fails.
Note that the computation cost of computing the Eigenvector Centrality measure for very large networks
is also rather expensive in computational time. As seen in Fig. 11 below, when the network size grow near
800,000 nodes, the computational cost of the most expensive Social method, i.e. Social 2 is already better
than that of computing one single time the Eigenvector Centrality measure. As contrast, in networks of sizes
of |๐| > 800,000 nodes, the computational cost of the Social 1 and the Social 0 is still negligible. As much
as a network of size |๐| > 1,000,000 nodes is still a very small network, the runtime is still less than 1
minute, thus it does not seem as the runtime in the Social methods is a real problem.
Fig. 11. Run time of different seeding methods for varying network sizes
VI. CONCLUSION
Many works that study information cascades in social networks, consider these cascades as a phenomenon
by which information virally spreads by its own force through the links of the network. Unlike the spread of
biological viruses that can be carried passively by agents and infect a significant portion of the network,
information cascades are usually much shorter and long cascades are rather rare [17], [15], [14]. These
results do not necessarily imply that social forces lost their importance but rather that people information
spread is more selective, and does not necessarily fit the use of an SIR model of virus spread.
We propose a new information spread model, in which agents, e.g., sales representative of a company,
communicate with network members, e.g., potential clients, and offer them a new product or service. The
probability that a client accepts such an offer is based on the acceptance levels of its neighbors.
Since contacting a client includes some financial cost (limiting the number of clients that can be
approached at once), the company has to select which members to approach and at what time, in order to
increase the total adoption rate in the network.
The proposed Latent Viral Marketing Model and its recommendation method for customer selection,
sees influential nodes, as nodes that are most likely to accept an offer at each period and thus influence
others.
In a large series of simulated experiments, we show that the proposed method increases the adoption rate
in 23%-153% (depending on the initial conditions), over the best-known method, which seeds the nodes by
their Eigenvector Centrality measure.
Having said that, it is important to note that the method is applicable to products that have a viral
characteristic. These are products or services where a substantial part of the purchasing decision is based on
social influence. In products or services where social forces are not important, it might still be better to use
the old method of selecting nodes that have high Eigenvalue Centrality measures.
The work contribution can be summarized along three different axes. First, we believe that the LVM
spread model better fits real-world scenarios of products adoption, where products spread relies on an effort
of a sales department, and seldom spread with no external force added. In these cases, this work directs the
sales personals, where and when to contact each possible customer. Second, the proposed model
demonstrates the importance and the high potential of a Scheduled Seeding approach, while restricting the
scenarios to the cases where this method is expected to be useful, as well as those where it is not. Third, we
offer a simple, yet a powerful method (by the SSH algorithm), that can be easily applied in disserve
situations of marketing of trendy product, where social forces are of high importance.
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